If you looking for feedback on your proposed algorithm (rather than code)...
I would probably take a similar approach, assuming that the array is large enough that a binary search is expected to be more efficient than creating an index of the first n characters of each string.
Keep in mind, though, that when you do the binary search for $beg you are getting information not only about the position of $beg, but also about $end. Each time you compare a string from the array to $beg you could also determine the relative location of $end to that string. Then, once you nail down the position of $beg you could potentially start searching for the exact position of $end using a much smaller search space. (The actual benefit achieved will depend on the positions of $beg and $end within the array.)
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